When you open the sharing dialogue to add or change who a site is shared with, you may see three different things displayed next to the email address(es) you have shared with
1. If you see a picture, then this email address has an associated Google account with a profile - the image comes from the profile.
2. If you see 'single head' icon, then this email address has an associated Google account without a profile
3. If you see a '3 head' icon, then either this email address has no associated Google account or is a Google group email address.
If you see 3, AND this is not a Google group email address then sharing will have no effect until that email address has an associated Google account. You may then need to delete and re-add the user. Also, modifying sharing at the page level using page level permissions will not work.
ALSO, changes to page level permissions will not 'stick' for non Google account addresses
Another icon you may see is
For what this means and the potential issue it represents, see Google Sites Sharing Issue